TITLE: Can a 3-manifold fiber over two different surfaces?
QUESTION [5 upvotes]: Let $M$ be a closed connected 3-manifold and let $\Sigma$ and $\Sigma'$ be two closed connected (not necessarily orientable) surfaces. Could there be  fiber bundles $\phi : M \to \Sigma$ and $\phi': M \to \Sigma'$ with $\Sigma \ncong \Sigma'$?

REPLY [4 votes]: Yes, it can. Consider $M=S^1 \times K^2$ where $K^2$ is the Klein bottle. This manifold fibers over both $T^2$ (since $K^2$ fibers over $S^1$) and over $K^2$. Of course, if the base of the fibration is required to have negative Euler characteristic, then $M$ cannot fiber over nonhomeomorphic surfaces.